[Théorèmes de type Baum–Katz pour un tableau de martingales]
Nous montrons la vitesse de convergence dans la loi des grand nombres pour un tableau de martingales. Les résultats étendent les théorèmes classiques de Baum et Katz (1965) [2] pour les sommes de variables aléatoires indépendantes et identiquement distribuées (i.i.d.). Ils améliorent un résultat de Ghosal et Chandra (1998) [6] pour des tableaux de martingales, et généralisent un résultat dʼAlsmeyer (1990) [1] pour une seule martingale. Comme application, nous obtenons un théorème nouveau concernant la vitesse de convergence pour des sommes de Cesàro de variables aléatoires identiquement distribuées.
We show convergence rates in the law of large numbers for martingale arrays. The results extend the classical theorems of Baum and Katz (1965) [2] for sums of independent and identically distributed (i.i.d.) random variables. They improve a result of Ghosal and Chandra (1998) [6] for martingale arrays, and generalize a result of Alsmeyer (1990) [1] for a single martingale. As an application, we obtain a new theorem about the convergence rate of Cesàro summation of identically distributed random variables.
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@article{CRMATH_2012__350_1-2_91_0,
author = {Hao, Shunli and Liu, Quansheng},
title = {Baum{\textendash}Katz type theorems for martingale arrays},
journal = {Comptes Rendus. Math\'ematique},
pages = {91--96},
publisher = {Elsevier},
volume = {350},
number = {1-2},
year = {2012},
doi = {10.1016/j.crma.2011.12.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.12.006/}
}
TY - JOUR AU - Hao, Shunli AU - Liu, Quansheng TI - Baum–Katz type theorems for martingale arrays JO - Comptes Rendus. Mathématique PY - 2012 SP - 91 EP - 96 VL - 350 IS - 1-2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.12.006/ DO - 10.1016/j.crma.2011.12.006 LA - en ID - CRMATH_2012__350_1-2_91_0 ER -
%0 Journal Article %A Hao, Shunli %A Liu, Quansheng %T Baum–Katz type theorems for martingale arrays %J Comptes Rendus. Mathématique %D 2012 %P 91-96 %V 350 %N 1-2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.12.006/ %R 10.1016/j.crma.2011.12.006 %G en %F CRMATH_2012__350_1-2_91_0
Hao, Shunli; Liu, Quansheng. Baum–Katz type theorems for martingale arrays. Comptes Rendus. Mathématique, Tome 350 (2012) no. 1-2, pp. 91-96. doi : 10.1016/j.crma.2011.12.006. http://www.numdam.org/articles/10.1016/j.crma.2011.12.006/
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